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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Wheeler-deWitt equation ： ウィキペディア英語版 Wheeler–DeWitt equation The Wheeler–DeWitt equation is an attempt to combine mathematically the ideas of quantum mechanics and general relativity, a step toward a theory of quantum gravity. In this approach, time plays no role in the equation, leading to the problem of time.〔https://medium.com/the-physics-arxiv-blog/d5d3dc850933〕 More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergmann-Komar "group" (which ''is'' the diffeomorphism group on-shell, but differs off-shell). Because of its connections with the low-energy effective field theory, it inherits all the problems of the naively quantized GR, and thus it cannot be used at multi-loop level, etc, at least not according to the current knowledge. The equation has not played a role in string theory thus far, since all properly defined and understood descriptions of string/M-theory deal with some fixed asymptotic conditions on the background. Thus, at infinity, the "right" choice of the time coordinate "t" is determined in every description, so there is a preferred definition of the Hamiltonian (with nonzero eigenvalues) to evolve states of the system forward in time. This avoids all the issues of the Wheeler-de Witt equation to dynamically generate a time dimension. But at the end, there could exist a Wheeler-de Witt style manner to describe the bulk dynamics of quantum theory of gravity. Some experts believe that this equation still holds the potential for understanding quantum gravity; however, decades after the equation was first written down, it has not brought physicists as clear results about quantum gravity as some of the results building on completely different approaches, such as string theory. == Motivation and background == In canonical gravity, spacetime is foliated into spacelike submanifolds. The three-metric (i.e., metric on the hypersurface) is $\backslash gamma\_$ and given by :$g\_\backslash ,\backslash mathrmx^\backslash ,\backslash mathrmx^=(-\backslash ,N^2+\backslash beta\_k\backslash beta^k)\backslash ,\backslash mathrmt^2+2\backslash beta\_k\backslash ,\backslash mathrmx^k\backslash ,\backslash mathrmt+\backslash gamma\_\backslash ,\backslash mathrmx^i\backslash ,\backslash mathrmx^j.$ In that equation the Roman indices run over the values 1, 2, 3 and the Greek indices run over the values 1, 2, 3, 4. The three-metric $\backslash gamma\_$ is the field, and we denote its conjugate momenta as $\backslash pi^$. The Hamiltonian is a constraint (characteristic of most relativistic systems) :$\backslash mathcal=\backslash frac\backslash pi^\backslash pi^-\backslash sqrt\backslash ,)$ and $G\_=(\backslash gamma\_\backslash gamma\_+\backslash gamma\_\backslash gamma\_-\backslash gamma\_\backslash gamma\_)$ is the Wheeler-DeWitt metric. Quantization "puts hats" on the momenta and field variables; that is, the functions of numbers in the classical case become operators that modify the state function in the quantum case. Thus we obtain the operator :$\backslash widehat\_\backslash widehat^\backslash widehat^-\backslash sqrt\backslash ,.$ Working in "position space", these operators are :$\backslash hat\_(t,x^k)\; \backslash to\; \backslash gamma\_(t,x^k)$ :$\backslash hat^(t,x^k)\; \backslash to\; -i\; \backslash frac.$ One can apply the operator to a general wave functional of the metric $\backslash widehat\{\backslash mathcal\{H\}\}\; \backslash Psi()\; =0$ where: :$\backslash Psi()\; =\; a\; +\; \backslash int\; \backslash psi(x)\; \backslash gamma(x)\; dx^3+\; \backslash int\backslash int\; \backslash psi(x,y)\backslash gamma(x)\backslash gamma(y)\; dx^3\; dy^3\; +...$ Which would give a set of constraints amongst the coefficients $\backslash psi(x,y,...)$. Which means the amplitudes for N gravitons at certain positions is related to the amplitudes for a different number of gravitons at different positions. Or one could use the two field formalism treating $\backslash omega(g)$ as an independent field so the wave function is $\backslash Psi()$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wheeler–DeWitt equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.